Logic in Computer Science : Modelling and Reasoning about Systems

439 Pages·2004·1.743 MB·English

by  Michael Huth, Mark Ryan

LOGIC IN COMPUTER SCIENCE
Modelling and Reasoning about Systems
MICHAEL HUTH
Department of Computing
Imperial College London, United Kingdom
MARK RYAN
School of Computer Science
University of Birmingham, United Kingdom
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Contents
Foreword to the first edition page ix
Preface to the second edition xi
Acknowledgements xiii
1 Propositional logic 1
1.1 Declarative sentences 2
1.2 Natural deduction 5
1.2.1 Rules for natural deduction 6
1.2.2 Derived rules 23
1.2.3 Natural deduction in summary 26
1.2.4 Provable equivalence 29
1.2.5 An aside: proof by contradiction 29
1.3 Propositional logic as a formal language 31
1.4 Semantics of propositional logic 36
1.4.1 The meaning of logical connectives 36
1.4.2 Mathematical induction 40
1.4.3 Soundness of propositional logic 45
1.4.4 Completeness of propositional logic 49
1.5 Normal forms 53
1.5.1 Semantic equivalence, satisfiability and validity 54
1.5.2 Conjunctive normal forms and validity 58
1.5.3 Horn clauses and satisfiability 65
1.6 SAT solvers 68
1.6.1 A linear solver 69
1.6.2Acubicsolver 72
1.7 Exercises 78
1.8 Bibliographic notes 91
2 Predicate logic 93
2.1 The need for a richer language 93
v
vi Contents
2.2 Predicate logic as a formal language 98
2.2.1 Terms 99
2.2.2 Formulas 100
2.2.3 Free and bound variables 102
2.2.4 Substitution 104
2.3 Proof theory of predicate logic 107
2.3.1 Natural deduction rules 107
2.3.2 Quantifier equivalences 117
2.4 Semantics of predicate logic 122
2.4.1 Models 123
2.4.2 Semantic entailment 129
2.4.3 The semantics of equality 130
2.5 Undecidability of predicate logic 131
2.6 Expressiveness of predicate logic 136
2.6.1 Existential second-order logic 139
2.6.2 Universal second-order logic 140
2.7 Micromodels of software 141
2.7.1 State machines 142
2.7.2 Alma – re-visited 146
2.7.3 A software micromodel 148
2.8 Exercises 157
2.9 Bibliographic notes 170
3 Verification by model checking 172
3.1 Motivation for verification 172
3.2 Linear-time temporal logic 175
3.2.1 Syntax of LTL 175
3.2.2 Semantics of LTL 178
3.2.3 Practical patterns of specifications 183
3.2.4 Important equivalences between LTL formulas 184
3.2.5 Adequate sets of connectives for LTL 186
3.3 Model checking: systems, tools, properties 187
3.3.1 Example: mutual exclusion 187
3.3.2TheNuSMVmodelchecker 191
3.3.3 Running NuSMV 194
3.3.4 Mutual exclusion revisited 195
3.3.5 The ferryman 199
3.3.6 The alternating bit protocol 203
3.4 Branching-time logic 207
3.4.1 Syntax of CTL 208
Contents vii
3.4.2 Semantics of CTL 211
3.4.3 Practical patterns of specifications 215
3.4.4 Important equivalences between CTL formulas 215
3.4.5 Adequate sets of CTL connectives 216
3.5 CTL* and the expressive powers of LTL and CTL 217
3.5.1 Boolean combinations of temporal formulas in CTL 220
3.5.2 Past operators in LTL 221
3.6 Model-checking algorithms 221
3.6.1 The CTL model-checking algorithm 222
3.6.2 CTL model checking with fairness 230
3.6.3 The LTL model-checking algorithm 232
3.7 The fixed-point characterisation of CTL 238
3.7.1 Monotone functions 240
3.7.2 The correctness of SATEG 242
3.7.3 The correctness of SATEU 243
3.8 Exercises 245
3.9 Bibliographic notes 254
4 Program verification 256
4.1 Why should we specify and verify code? 257
4.2 A framework for software verification 258
4.2.1 A core programming language 259
4.2.2 Hoare triples 262
4.2.3 Partial and total correctness 265
4.2.4 Program variables and logical variables 268
4.3 Proof calculus for partial correctness 269
4.3.1 Proof rules 269
4.3.2 Proof tableaux 273
4.3.3 A case study: minimal-sum section 287
4.4 Proof calculus for total correctness 292
4.5 Programming by contract 296
4.6 Exercises 299
4.7 Bibliographic notes 304
5 Modal logics and agents 306
5.1 Modes of truth 306
5.2 Basic modal logic 307
5.2.1 Syntax 307
5.2.2 Semantics 308
5.3 Logic engineering 316
5.3.1 The stock of valid formulas 317
viii Contents
5.3.2 Important properties of the accessibility relation 320
5.3.3 Correspondence theory 322
5.3.4 Some modal logics 326
5.4 Natural deduction 328
5.5 Reasoning about knowledge in a multi-agent system 331
5.5.1 Some examples 332
5.5.2 The modal logic KT45n 335
5.5.3 Natural deduction for KT45n 339
5.5.4 Formalising the examples 342
5.6 Exercises 350
5.7 Bibliographic notes 356
6 Binary decision diagrams 358
6.1 Representing boolean functions 358
6.1.1 Propositional formulas and truth tables 359
6.1.2 Binary decision diagrams 361
6.1.3 Ordered BDDs 366
6.2 Algorithms for reduced OBDDs 372
6.2.1 The algorithm reduce 372
6.2.2 The algorithm apply 373
6.2.3 The algorithm restrict 377
6.2.4 The algorithm exists 377
6.2.5 Assessment of OBDDs 380
6.3 Symbolic model checking 382
6.3.1 Representing subsets of the set of states 383
6.3.2 Representing the transition relation 385
6.3.3 Implementing the functions pre and pre 387
∃ ∀
6.3.4 Synthesising OBDDs 387
6.4 A relational mu-calculus 390
6.4.1 Syntax and semantics 390
6.4.2 Coding CTL models and specifications 393
6.5 Exercises 398
6.6 Bibliographic notes 413
Bibliography 414
Index 418
Foreword to the first edition
by
Edmund M. Clarke
FORE Systems Professor of Computer Science
Carnegie Mellon University
Pittsburgh, PA
Formal methods have finally come of age! Specification languages, theorem
provers, and model checkers are beginning to be used routinely in industry.
Mathematical logic is basic to all of these techniques. Until now textbooks
on logic for computer scientists have not kept pace with the development
of tools for hardware and software specification and verification. For exam-
ple, in spite of the success of model checking in verifying sequential circuit
designs and communication protocols, until now I did not know of a sin-
gle text, suitable for undergraduate and beginning graduate students, that
attempts to explain how this technique works. As a result, this material is
rarelytaughttocomputerscientistsandelectricalengineerswhowillneedto
use it as part of their jobs in the near future. Instead, engineers avoid using
formalmethodsinsituationswherethemethodswouldbeofgenuinebenefit
or complain that the concepts and notation used by the tools are compli-
cated and unnatural. This is unfortunate since the underlying mathematics
is generally quite simple, certainly no more difficult than the concepts from
mathematical analysis that every calculus student is expected to learn.
Logic in Computer Science by Huth and Ryan is an exceptional book.
I was amazed when I looked through it for the first time. In addition to
propositional and predicate logic, it has a particularly thorough treatment
of temporal logic and model checking. In fact, the book is quite remarkable
in how much of this material it is able to cover: linear and branching time
temporal logic, explicit state model checking, fairness, the basic fixpoint
ix
x Foreword to the first edition
theorems for computation tree logic (CTL), even binary decision diagrams
and symbolic model checking. Moreover, this material is presented at a level
that is accessible to undergraduate and beginning graduate students. Nu-
merous problems and examples are provided to help students master the
material in the book. Since both Huth and Ryan are active researchers in
logics of programs and program verification, they write with considerable
authority.
In summary, the material in this book is up-to-date, practical, and ele-
gantly presented. The book is a wonderful example of what a modern text
on logic for computer science should be like. I recommend it to the reader
with greatest enthusiasm and predict that the book will be an enormous
success.
(This foreword is re-printed in the second edition with its author’s permis-
sion.)
Preface to the second edition
Our motivation for (re)writing this book
One of the leitmotifs of writing the first edition of our book was the obser-
vation that most logics used in the design, specification and verification of
computer systems fundamentally deal with a satisfaction relation
M (cid:1) φ
where M is some sort of situation or model of a system, and φ is a specifi-
cation, a formula of that logic, expressing what should be true in situation
M. At the heart of this set-up is that one can often specify and implement
algorithms for computing (cid:1). We developed this theme for propositional,
first-order, temporal, modal, and program logics. Based on the encourag-
ing feedback received from five continents we are pleased to hereby present
the second edition of this text which means to preserve and improve on the
original intent of the first edition.
What’s new and what’s gone
Chapter 1 now discusses the design, correctness, and complexity of a SAT
solver (a marking algorithm similar to St˚almarck’s method [SS90]) for full
propositional logic.
Chapter 2 now contains basic results from model theory (Compactness
Theorem and L¨owenheim–Skolem Theorem); a section on the transitive clo-
sure and the expressiveness of existential and universal second-order logic;
andasectionontheuseoftheobjectmodellinglanguageAlloyanditsanal-
yserforspecifyingandexploringunder-specifiedfirst-orderlogicmodelswith
respect to properties written in first-order logic with transitive closure. The
Alloy language is executable which makes such exploration interactive and
formal.
xi
xii Preface to the second edition
Chapter 3 has been completely restructured. It now begins with a discus-
sion of linear-time temporal logic; features the open-source NuSMV model-
checking tool throughout; and includes a discussion on planning problems,
more material on the expressiveness of temporal logics, and new modelling
examples.
Chapter 4 contains more material on total correctness proofs and a new
section on the programming-by-contract paradigm of verifying program cor-
rectness.
Chapters 5 and 6 have also been revised, with many small alterations and
corrections.
The interdependence of chapters and prerequisites
The book requires that students know the basics of elementary arithmetic
and naive set theoretic concepts and notation. The core material of Chap-
ter 1 (everything except Sections 1.4.3 to 1.6.2) is essential for all of the
chapters that follow. Other than that, only Chapter 6 depends on Chapter 3
and a basic understanding of the static scoping rules covered in Chapter 2 –
although one may easily cover Sections 6.1 and 6.2 without having done
Chapter 3 at all. Roughly, the interdependence diagram of chapters is
1
2 3 4 5
6
WWW page
This book is supported by a Web page, which contains a list of errata;
text files for all the program code; ancillary technical material and links;
all the figures; an interactive tutor based on multiple-choice questions;
and details of how instructors can obtain the solutions to exercises in
this book which are marked with a ∗. The URL for the book’s page
is www.cs.bham.ac.uk/research/lics/. See also www.cambridge.org/
052154310x